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3. Draw line DE, cutting AB in F.

4. From D, with radius DA, describe arc AB.

5. From E, with radius EF, describe an arc, cutting the arc AB in G and H.

6. From C, with line GH as radius, cut the semicircle in 1. 7. Draw line B 1; it is a second side of the nonagon.

8. Bisect B 1, and obtain O, the centre of the circle.

9. Mark off, on the circumference, the divisions 12, 23, &c., equal to B 1. Join 12, 23, &c., and a nonagon is constructed on the given line AB.

Problem 76.

To construct a regular decagon on a given line AB.

1. Produce the line AB, and from B, with radius BA, describe a semicircle, cutting it in C.

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2. From A, with radius AB, describe an arc, cutting the semicircle in D, and bisect AB in E (Pr. 1).

3. From B, with radius BE, describe an arc, cutting arc BD in F.

4. Draw line EF.

5. From C, with radius EF, cut the semicircle in 1; then B1 is a second side of the decagon.

6. Bisect B 1, and obtain O, the centre of the circle.

7. Mark off, on the circumference, the divisions 1 2, 23, &c, equal to B 1. Join 1 2, 2 3, &c., and a decagon is constructed on the given line AB.

Problem 77.

To construct a regular un-decagon on a given line AB.

1. Produce the line AB, and from B, with radius BA, describe an arc, cutting the produced line AB in C, and being produced below A.

2. From A, with the same radius, describe an arc, cutting the first arc in D and E.

3. Draw line DE, bisecting AB.

2

5

K

E

4. From B, with half BA as radius, describe an arc cutting BD in G.

5. Bisect the arc BE in H; and draw AG, AH, cutting ED in K and L.

6. From C, with radius AL, cut off C 1 on the semicircle.

7. Draw line B1; it is a second side of the un-decagon.

8. Bisect B 1, and obtain O, the centre of the circle.

9. Mark off, on the circumference, the divisions 1 2, 23, &c., equal to B 1. Join 1 2, 2 3, &c., and an un-decagon is constructed on the given line AB.

Problem 78.

To complete a regular polygon, its two sides AB, BC being

given.

1. Bisect the lines AB, BC by perpendiculars meeting at O (Pr. 1).

E

2. With centre 0, and radius OA, describe the circle.
3. From A, mark off the distance AB to DE, &c. Join
AD, DE, EF, &c., and a regular polygon will be com-
pleted—in this case a hexagon.

Problem 79.

To construct a regular hexagon, its diameter AB being given. 1. Bisect AB in C (Pr. 1).

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2. Through the point A draw a line DE perpendicular to AB (Pr. 2).

3. On CA, as an altitude, construct an equilateral triangle, having its vertical angle at C (Pr. 19).

4. From C, with radius CE or CD, describe a circle.

5. From point E, mark off the distance ED to FG, &c. Join EF, FG, &c., and a regular hexagon will be con structed, having the given diameter AB.

SECTION VI.-ELLIPSES, &c.

DEFINITIONS.

In order to understand the following definitions clearly, we must refer to that SOLID which is called a CONE.

1. A cone is a solid figure, the base of which is a circle, but which tapers to a point from the base upward. Ex. ABC

A

NOTE 1.-A straight line drawn from the centre of the base to the apex (or summit) is called its axis. Ex. AD

NOTE 2. When the apex is perpendicular to the base, the cone is said to be a right cone.

NOTE 3.-When the axis is not perpendicular to the base, the cone is said to be an oblique cone.

NOTE 4.-If a right cone be cut in two parts by a plane parallel

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